*Solution Methodology*

For the numerical evaluation of the flow equations in the plume region, the finite difference scheme is implemented. The constitutive equations in discretized forms are given as below:

$$\mathcal{W}\_{i,j} = \frac{1}{\left(\Delta Z + Z\_j\right)} \left\{ Z\_j \mathcal{W}\_{i-1,j} - Z\_j \left( \mathcal{U}\_{i,j} - \mathcal{U}\_{i,j-1} \right) \frac{\Delta Z}{\Delta X} + \frac{Z\_j^2}{8X\_i} \left( \mathcal{U}\_{i+1,j} - \mathcal{U}\_{i-1,j} \right) - \frac{3}{4} \Delta Z Z\_j \mathcal{U}\_{i,j} \right\} \tag{43}$$

$$\begin{cases} \left[\frac{1}{2}\Delta Z\left(W\_{i,j} - \frac{1}{4}Z\_{j}\mathbb{I}I\_{i,j} - \frac{1}{2\overline{Z\_{j}}}\right) + 1\right] \mathcal{U}\_{i-1,j} \\ \quad + \quad \left\{\left[\Delta Z^{2}\left(-X\_{i}\mathbb{I}I\_{i,j}\frac{1}{\Delta X} - \frac{1}{2}\mathbb{I}I\_{i,j} - X\_{i}^{1/2}\mathcal{M}\right) - 2\right] \mathcal{U}\_{\text{I}1,j} \\ \quad + \quad \left\{\left[-\frac{1}{2}\Delta Z\left(\mathcal{W}\_{i,j} - \frac{1}{4}Z\_{j}\mathbb{I}I\_{i,j} - \frac{1}{2\Sigma\_{j}}\right) + 1\right] \mathcal{U}\_{i+\mathbf{1},j} \\ \quad = \Delta Z^{2}\left(\overline{\Theta}\_{i,j} + \overline{\varphi}\_{i,j} - X\_{i}\mathbb{I}I\_{i,j}U\_{i,j-1}\left(\frac{1}{\Delta X}\right)\right) \end{cases} \tag{44}$$

$$\begin{split} \left[ \frac{1}{2} \Delta \mathcal{L} \Big( \mathcal{W}\_{i,j} - \frac{1}{4} \mathcal{Z}\_{j} \mathcal{U}\_{i,j} - \frac{1}{2 \mathcal{V}\_{i} \mathcal{Z}\_{j}} \Big) + \mathrm{Nb} \Big\{ \overline{\mathcal{Q}}\_{i+1,j} - \overline{\mathcal{q}}\_{i-1,j} \Big\} + \mathrm{Nt} \Big\{ \overline{\mathcal{Q}}\_{i+1,j} - \overline{\mathcal{Q}}\_{i-1,j} \Big\} \right] \\ \quad + \mathrm{Tr} \Big[ \overline{\Theta}\_{i-1,j} + \Big\{ -\Delta \mathcal{Z}^{2} \Big[ \frac{\mathcal{X}\_{i}}{\Delta \mathcal{X}} \mathcal{U}\_{i,j} - \mathcal{Q} \Big] - \frac{2}{\mathrm{Pr}} \Big{\overline{\Theta}\_{i,j}} \Big{\} \\ \quad + \Big\{ - \Big[ \frac{1}{2} \Delta \mathcal{Z} \Big( \mathcal{W}\_{i,j} - \frac{1}{4} \mathcal{Z}\_{j} \mathcal{U}\_{i,j} - \frac{1}{2 \mathcal{P}\_{i} \mathcal{Z}\_{i}} \Big{)} + \mathrm{Nb} \frac{1}{4} \Big{\overline{\mathcal{Q}}\_{i+1,j} - \overline{\mathcal{q}}\_{i-1,j} \Big{)} \\ \quad + \mathrm{Nt} \frac{1}{4} \Big{(} \overline{\mathcal{Q}}\_{i+1,j} - \overline{\mathcal{Q}}\_{i-1,j} \Big{)} \Big{]} + \frac{1}{\mathrm{Pr}} \Big{\} \overline{\Theta}\_{i+1,j} = -X\_{i} \boldsymbol{I} I\_{i,j} \frac{\Delta \mathcal{X}^{2}}{\Delta \mathcal{X}} \overline{\theta}\_{i,j-1} \\ \quad + \Big{(} \frac{1}{2} \Delta$$

With boundary conditions:

$$\begin{aligned} \mathcal{W}\_{i,j} = 0, \quad \mathcal{U}\_{i+1,j} = \mathcal{U}\_{i-1,j}, \quad & \overline{\theta}\_{i+1,j} = \overline{\theta}\_{i-1,j}, \; \overline{\varphi}\_{i+1,j} = \overline{\varphi}\_{i-1,j} \\ & \overline{\varphi}\_{i,j} = 1, \quad & \overline{\theta}\_{i,j} = 1 \quad \text{at } Z\_{j} = 0, \\ \mathcal{U}\_{i,j} \to 0 \, \, \prime \qquad & \overline{\varphi}\_{i,j} \to 0 \, \, \prime \qquad \qquad \qquad \qquad \text{as} \quad Z\_{j} \to \infty. \end{aligned} \tag{47}$$
